The Power of Many Samples in Query Complexity

TL;DR

This paper investigates whether access to infinitely many samples simplifies the task of distinguishing distributions in query complexity, and finds that it does not, establishing a lower bound related to the original complexity.

Contribution

It proves that even with unlimited samples, distinguishing certain distributions remains as hard as the original query complexity, and derives a new lower bound for composed functions.

Findings

Infinite sampling does not reduce query complexity for certain distributions.

Established a lower bound for the randomized query complexity of composed functions.

Connected fractional block sensitivity to query complexity in composition.

Abstract

The randomized query complexity $R(f)$ of a boolean function $f\colon\{0,1\}^n\to\{0,1\}$ is famously characterized (via Yao's minimax) by the least number of queries needed to distinguish a distribution $D_0$ over $0$-inputs from a distribution $D_1$ over $1$-inputs, maximized over all pairs $(D_0,D_1)$. We ask: Does this task become easier if we allow query access to infinitely many samples from either $D_0$ or $D_1$? We show the answer is no: There exists a hard pair $(D_0,D_1)$ such that distinguishing $D_0^\infty$ from $D_1^\infty$ requires $\Theta(R(f))$ many queries. As an application, we show that for any composed function $f\circ g$ we have $R(f\circ g) \geq \Omega(\mathrm{fbs}(f)R(g))$ where $\mathrm{fbs}$ denotes fractional block sensitivity.